Question

There are two similar games played for Chinese New Year and Vietnamese New Year. In the Chinese version, fair dice with numbers 1, 2, 3, 4, 5, and 6 are used, along with a board with those numbers. In the Vietnamese version, fair dice with pictures of a gourd, fish, rooster, crab, crayfish, and deer are used. The board has those six objects on it, also. We will play with bets being $1. The player places a bet on a number or object. The "house" rolls three dice. If none of the dice show the number or object that was bet, the house keeps the $1 bet. If one of the dice shows the number or object bet (and the other two do not show it), the player gets back his or her $1 bet, plus $1 profit. If two of the dice show the number or object bet (and the third die does not show it), the player gets back his or her $1 bet, plus $2 profit. If all three dice show the number or object bet, the player gets back his or her $1 bet, plus $3 profit. Let X = number of matches and Y = profit per game.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. List the values that Y may take on. Then, construct one PDF table that includes both X and Y and their probabilities.

e. Calculate the average expected matches over the long run of playing this game for the player.

f. Calculate the average expected earnings over the long run of playing this game for the player

g. Determine who has the advantage, the player or the house.

Answer 1

The random variable X represents the number of dice that match a player's bet in the game. The values of X can be 0, 1, 2, or 3, with a Binomial distribution X ~ B(3, 1/6). The average expected number of matches and profit can be calculated from the PDF table and are used to assess the house's advantage in the game.

Step-by-step explanation:

Let X be the random variable representing the number of matches (between the dice roll and the player's bet) in the described games. The values that X may take on are 0, 1, 2, or 3, with respective probabilities based on the roll of three fair six-sided dice or pictorial dice.

Distribution of X

X follows a Binomial distribution, since there are a fixed number of independent trials (three dice rolls), two outcomes (match or no match), and the probability of a match is the same for each roll. The probability of a match is 1/6, and the probability of no match is 5/6. Therefore, the distribution is X ~ B(3, 1/6).

Values for Y and PDF Table

The values that Y, the profit per game, may take on are -1, 1, 2, or 3 dollars. The PDF table is constructed by calculating the probabilities for 0, 1, 2, or 3 matches and associating them with the corresponding profits.

Expected Number of Matches and Profit

The average expected number of matches (E(X)) over the long run is calculated using the formula for the expected value of a binomial distribution. Similarly, the average expected profit (E(Y)) can be calculated using the expected number of matches and associated profit values.

House Advantage

To determine who has the advantage in this game, we calculate the expected profit for the player and compare it to the amount the house is expected to retain. An expected profit less than zero indicates a house advantage.